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Liminf to limsup transformation question

  1. Jan 18, 2009 #1
    y_n is a bounded sequence

    liminf(-y_n)=-limsup(y_n)
    n->infinity

    i cant understand how it happens??
     
  2. jcsd
  3. Jan 18, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Multiply as sequence of number reverses the direction. If, for example, the sequence were increasing, say 1, 2, 3, 4, etc., then multiplying by -1 changes it to -1, -2, -3, -4, etc. which is decreasing.

    The sequence 1/2, 0, 2/3, 0, 3/4, ..., does not converge but has one subsequence, 0, 0, 0, 0,..., that converges to 0 and another, 1/2, 2/3, 3/4, ..., that converges to 1. For this sequence liminf= 0 and limsup= 1. Multiplying each term by -1 gives -1/2, 0, -2/3, 0, -3/4, ... which has one sequence, 0, 0, 0, ..., that converges to 0 and another, -1/2, -2/3, -3/4, ..., that converges to -1. For this sequence, liminf= -1 and limsup= 0.
     
  4. Jan 18, 2009 #3
    thanks
     
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